IPLAAIC FLOW (Croy: ) PIR0 EPTIES
CIF EVC YELLOW 1131PCI1 !PLY
I• 01D
PLATES UNINT CONSTANT
S1-11A1? STRESS
Information IReviewed and Reaffirmed
October 1960
No. 1324
111111 11111111111111111
M111111111
1111
11111
14
111
FOREST PRODUCTS LABORATORY
NITED STATES DEPARTMENT OF AGRICULTURE
FOREST SERVICE
MADISON 5, WISCONSIN
n Cooperation with the University of Wisconsin
PLASTIC FLOW (CREEP) PROPERTIES OF TWO YELLOW BIRCH
PLYWOOD PLATES UNDER CONSTANT SHEAR STRESSba
By
C. B. NORRIS, Engineer
and
W. J. 'COMERS, Engineer
Forest Products Laboratory, l Forest Service
U. S. Department of Agriculture
Summary
This report presents the results of tests made on two yellow birch plywood plates subjected to constant shear stresses to determine the plastic flow or creep properties of the material. The plates were of apecial 1:3:1 construction, in which the core consisted of three parallelply laminations. A. mathematical analysis of a relationship between deflection ) load) and time, satisfying the data obtained from tests, is
also given. The tests were made by applying load (dead weights) perpendicular to the plane of a square plate at diagonally opposite corners, with supports at the other two corners. This leading has been
demonstrated to subject the plate to shearing stress.. The plastic
deflection-time curves obtained are parabolic. By testing a single
specimen at various loads, a family of plastic deflection-time curves
is obtained, one curve for each load. If the data are replotted in a
family of plastic deflection-load curves, one curve for each time, the
curves are again parabolic.
1
This report is one of a series of progress reports prepared by the
Forest Products Laboratory relati ng to the use of wood in aircraft.
Results here reported are preliminary and may be revised • as additional data become available.
—
2
—Original report dated October 1943.
Maintained at Madison, Wis., in cooperation with the University of
Wisconsin.
14
—
Forest Products Laboratory Report No. 1301.
Report No. 1324
The equation is derived from the assumption that:
a . KEIT1 (dfln
dt
and is
1
2
=
/
m + n km
a mi-nin+n m-i-n4.
2
n
h
2Kh
1
n
n
m ni
Ewa,
where a is shear stress (pounds per square inch)
• is plastic shear strain
w is plastic deflection (inches) at gage points with respect to
center of plate
P is load (pounds)
t is time (hours)
a is 1/2 the distance (inches) between adjacent gage points
h is thickness of plate (inches)
K, m4 and n are constants of the material
(See Appendix for derivation).
Introduction
Creep or flow of materials deals with their plastic or inelastic behavior. Some solids, such as asphalt, flow appreciably at room temperatures under small loads; at sufficiently high temperatures nearly all
metals will creep under stress. The subject of creep has been under investigation for many years for some of the widely used metals of construction. Wood is known to exhibit some plastic behavior, but very few
data have been obtained on the subject. The work described in this report is part of the program in progress at the Forest Products Laboratory on the relation of the strength properties of wood. and plywood to
the duration of stress.
Report No. 1324
-2-
Material Tested
The two test specimens were cut from two panels of yellow birch plywood
each consisting of five 1/16-inch plies arranged with the grain of the
three central plies parallel to afford 1:3:1 construction. Each ply
was a single piece of veneer. The thickness of the glue lines, the
weight of the glue, and creep in the glue are assumed to be negligible
throughout the calculations. The panels were received in a shipment
of Tego-bonded aircraft plywood. The average specific gravity of the
material tested was 0.64 based on the weight when oven dry and the volume at t es t. All tests were conducted under controlled conditions of
80° F. and 50 percent relative humidity resulting in approximately 9
percent moisture content in the wood. The specimens had dimensions of
11.0 by 11.0 by 0.300 inches.
Method of Test
Tests were made using the apparatus shown in figure 1. The specimens
were placed so as to apply four equal loads at the corners of the square
plate; two, at the extremity of one diagonal, acting downward (dead
weights) and two, at the extremity of the other diagonal, acting upward
(supports)42 Small metal corner pieces were attached to the specimen
so that the loads could be applied directly at the corners of the square.
The deflection to be considered is the vertical displacement of a point
on a diagonal with respect to the center of the specimen. With the arrangement used, as shown in figure 1, the mean upward displacement of
two symmetrically located points on one diagonal is automatically added
to the mean downward displacement of two similar points on the other diagonal. One half the dial gauge reading is thus the average of the vertical displacements of the four points with respect to the center of the
specimen.
Weights were hung from the ends of one diagonal and deflection readings
taken at approximately 1, 3, 6, 12, 18, 24 1 and 30 minutes and 1, 3 1 6,
9, 12, 16, and 24 hours. The plates were alternated each 24-hour period
and, in addition, each plate was loaded alternately on 'each diagonal.
This report is based on 27 such loadings; 15 on one specimen and 12 on
the other. One loading was with 10 pounds (5 pounds at each end of the
diagonal, 11 with 14 pounds ) 9 with 20 pounds, and 6 with 24 pounds.
Loads less than 10 pounds were not used, as deflection increments due to
2Forest Products Laboratory Report No. 1301.
Report No. 1324
_3..
such loads are too small to read with sufficient accuracy, and loads
over 24 pounds . were not used, as direct stresses due to bending of the
plate become appreciable.
The recorded deflections are with respect to a zero established after
the specimen was in place with the metal corners, hooks, and dial supports attached and the analysis assumes that all creep Is due to the
applied loads and none to the weight of the plate and the attached
apparatus.
Analysis of Results
The time-deflection readings recorded during the tests were plotted on
logarithmic graph paper. The resulting curves were of the shape shown
in figure 2, curve A. The points of the middle curve are the same as
those of Curve A with a correction term c added to each deflection
reading which is the difference between the deflection at the observed
zero time and the actual zero time. This difference is due to the difficulty of placing the weights on the specimen at any particular instant. By solving for this correction (see appendix), and replotting,
the points are close to a straight line. The equation of this line is
of the form w = atb where a and b are constants for the particular curve
drawn. By testing the specimens several times at each load it was possible to determine average values of a and b for each load, in the equation w = atb , which are more accurate than the values determined from
any single test.
It was found that b
— did not vary appreciably with the load, so the val.ues of b were averaged for all tests made, and arithmetically weighted
according to the smoothness of the curve obtained from the test data.
A value of a was chosen for each load which caused the plotted points
of the equation to agree most closely with all of the test points for
that load. These values of a were plotted against the corresponding
loads on logarithmic graph paper (fig. 3). A straight line averaging
these points was drawn and its equation was determined. The value of a
in terms of the load, determined by this method, substituted in the
equation w atb yields equation (16) of the appendix,
0.0007 P°' 54 t°° 22
Figure 4 shows the comparison between the theoretical (equation (16) of
the appendix) and average time-deflection curves plotted from data obtained from the two yellow birch plywood plates. The solid lines represent the theoretical curves. The plotted points result from using
the average values of b from the equation w atb for all the tests
Report No. 1324
made and the value of a for each load determined as indicated. From this
figure, it can be seen that for the four loads used, one test curve was
coincident with the theoretical, two were slightly lower, and one was
above.
It was realized that the plastic constants of one piece of plywood might
be quite different from those of another. It was decided, therefore, to
make a number of tests upon a single specimen in the preliminary study
rather than to make a single test upon a number of specimens. It was
assumed, because of the small loads used, that the plastic constants
would not be materially changed by the treatment endured by the specimen
during the several tests.
It has been found from other studies6 that repeated and reversed stresses
with loads greater in proportion to dimensions of specimens do have an
effect on the modulus of elasticity of wood and plywood in bending and
compression. It is also safe to assume that repeated and reversed
stresses may affect the values of the plastic flow constants, even though
24 hours elapsed between tests of the P lates. This effect may account in
part for the discrepancy between the theoretical and experimental points
shown in figure 4.
The discrepancy may also be due in part to the additional weight of the
testing apparatus which could not be taken into account in the theory and
to small inaccuracies in applying the loads exactly at the corners of the
plates. The test points for the 10 pound load are for one test only and
are not the averages of several tests as are the other load-deflection
points of figure 4.
The middle curve of figure 2 shows the actual test points taken during a
test of a plate under a 14-pound load. The recorded deflections have
been corrected for the zero ' time error and show very good agreement with
the theoretical curve of figure 2 which represents equation (16) of the
appendix for a 14-pound load.
Table 1 lists the 27. tests with the equations of the curves as first
plotted and the equations using a common exponent (the average of all
tests). Also given are arithmetic weights used for each equation in
calculating the average constant for each load.
The formulas in the appendix have been derived for solid wood plates with
the surface in the LT plane . (longitudinal-tangential). For rotary-cut
veneer that is made into plywood, the surface of each ply is in the LT
–Forest Products Laboratory Report No. 1320.
Report No. 1324
plane also, so that in this test (neglecting the effect of the glue) the
same formulas apply in rotary-cut plywood plates as in flat-sawn wood
plates. All data from these tests give the effect of flow in only the
LT plane of shear, but it is expected that the flow in the LR
(longitudinal-radial) plane will bp of somewhat the same magnitude.
Conclusions
The data show that flat yellow birch plywood plates of 1:3:1 construc
tion, when subjected to constant shear stress in the LT plane exhibit
plastic flow at a rate which decreases with time. The plastic
deformation-time curve is parabolic in form and concave downward. It is
also shown that the plastic deformation at any given time is a parabolic
function of the load such that the plastic deformation-load curve is
concave downward.
From the limited number of loadings on only two plates of one material,
it is believed that the tentative formula presented will give reasonable
values of creep or plastic flow in shear, but it should not be assumed
that other stresses on wood, such as compression or tension, will produce plastic flow according to the theory presented. Tests were not
continued for more than 24 hours on any specimen, and while it is assumed that the theory applies to long-time tests, no such test data are
yet available.
It appears that the assumption
a=
, m ideNn
nE k--1
dt
is a reasonable relation between the shear stress and the plastic shear
strain.
Report No, 1324
-6-
Appendix
Notations used .in calculations:
X, Y, and Z = choice of axes shown in figure IB.
w = plastic deflection, of a point on the surface of a square plate
with respect to the center.
w 1 = deflection of a point on the surface of a square plate with
respect to the center.
= shear strain at'a distance z from center plane of plate.
e = shear strain at the surface of the plate.
1
. plate thickness.
= distance from center of plate to projection of gage point on
the x or x. axis.
P
= total load on plate pounds).
= time (hours).
m, and n constants of the wood.
. shear stress at central plane of the plate.
= shear stress at surface of the plate.
= plastic deformation at a distance z from the central plane of
the plate.
= plastic deformation at the surface of the plate.
Sh-- at central plane of the plate.
dt
de l
at surface of the plate.
u1 =
twisting moment.
Mxy
a-
Report No. 1324
-7-
Calculations
From the theory of thin plates, it may be found1):- that for the case here
considered the shear strain component is:
e
eur
=
ax y
Assuming that e
1 :30
w
rel
I
is
37
independent of X and and that w1 = ,0 when x = 0 and
where z =
(1)
An elastic surface of this kind is obtained if a square plate is supported and loaded as shown in fi gure 1.2.
In this
case*t,
(2)
P=414
xy
Since the load P is a constant and the deflectionw1 is measured at var.
bus time intervals, it is assumed that the elastic deformation can be
neglected, if the first deflection reading is taken just alter the load
is applied. It is also assumed that lines normal to the original surface of the plate will remain straight for plastic deformations as Nell
as elastic. Equation (1) then applies to the plastic deformations!.-, and
w =
el xy
and
E
=
2e, –
(4)
Forest Products Laboratory Report No. 1301.
--frimoshenkos S. Theory of Plates and Shells. P. 91.
Forest Products Laboratory Report No,. 1301.
10
--"MacCulloughp G. E. An Experimental and Analytical Investigation of
Creep in sending. Applied Mechanics, April-3une 1933.
Report NO. 1324
-8-
de
pu z
h
Let u = dt = 2
-
--e
h
(5)
-
where 1
u -- value of u at the surface of the plate.
(6)
Assume a Kern un
Substituting (4) and (5) in (6)
= K (2e12u1 h)n
mn
1
=
Then
+n
o
where a 1 =
=
1
Now
E
mn
1
K€1
(7)
12
h
2
M.2
xy
n
z )111
(8)
az dz
Substituting (7) in (8)
h
zm+n+ldz
= 2a 1
0
h
o h2
M = 1 m +1 n + 2
xy
(m + n + 2)
h2
-Davis, E. A, Creep of Metals at High Temperature in Bending. Journal
of Applied Mechanics. March 1938.
1 '
Products Laboratory Report No. 1312.
Report No. 1324
Substituting (2) in (9)
P (m + n + 2)
v 2
2h
(io)
From equation (7)
=KE
m
1
u
n
1
e(--)
1 dt
n
(n)
Substituting (11) in (10)
de n
m
1.
Ke (---)
1 dt
(m n
2
2h
de
1 = P m + n + 2)
1 ""
2
E 1 dt
2Kh
Or
E1
de
1=
P (m + n+
2Kh2
1
TT
dt
Since P does not vary with t
1
(m + n + 2)
2
2Kh
n
t+c
If e = 0 when t 0 then C=
Then
+1)
Report No. 1324
m+n
[P(m+n+2)
2
21th
-10-
1
n
m + n m + n)
(12)
Assuming that equation (3)) applies to the plastic case, and substituting
it in (12) where x
m+n
[P
1
,1m"--1---- n
+ n + 2)
m
+n
4
a
or
n
+ n m + n
W=
+ n + 2
2Kh 2
1
n
1
m+n E7T–ff
n
t
P
This is the equation that has been used.
Sample calculation from data obtained during the test of a plywood plate
under a load P of 14 pounds:
Curve A, figure 2, is a plotting of deflection readings taken during the
test. This curve is of the form w = at b + c where c is the difference
in deflections at the observed zero time and the actual zero time.
tl
2
= 0.2
w 1 = 0.00099
= .2.0
.00227
t = 20.0
w W3 •w 2
1 3
w + w
3
a.
x0440
0.00094
By adding this constant to all readings, and replotting (middle curve of
fig. 2), a straight line can be drawn through the test points. Solving
for the function:
w = 0.00275 1
Report No. 1324
.221
-11-
The average slope of the lines from all tests was found to be 0.22. The
following equation results from drawing the best straight line of this
slope through the test points of all the tests made with this load.
w
15)
0.005 4 3 tU.22
2
From equation (13), since P does not vary with t ) the value of m n
= 0.22.
Plotting w against P when t = 1 on logarithmic P aper for the various
tests (fig. 3), it is possible to draw a straight line through the
points, the slope of which yields the value of .174-7--E = 0.54.
Thus m = 1.44
= 0.4(6
Knowing these values, it is possible to solve for K which equals
1.433 x 101°
The final equation reduces to
= O0
0.0
P
Report No. 1324
0.54 0.22
(16)
-.12-
.2-18
Table 1.--Results of tests to determine 'lastic flow (creep
properties ofjellow birch plywood of 1:3:1
construction under constant shear stress
••
:Arith-:Weighted
: metic: average
.
•
:weight:of a for
•
:
:b = 0.22
:
:Individual curves as plotted: a for :
:
•
:curves with:
:
:
:
: b = 0.22 :
b
:
a
:
_
•
.......0 ......: ------ :____:
:
•
:
:
•:
•: .
:
Lb.:
:
:
.
.
: 0.0026 : 7 : 0.0026
0.197
0.0029
10 :
1 :
C
8 : .0028
.0027 :
:
.248
:
2 :
C
14 :
.0023
8 : .0028
.0029 :
:
.118
.0056
:
3 : C : 14 :
•
.0025 : 6 : .0028
.265
:
.0021
14 :
4 : C
Test:Speci-:Load:
•
No.: men :
5 :
14
14 :
16 :
A : 14
A : 14 :
A : 14:
A : 14 :
A
14 :
A : 14
C : 20 :
20 :
C
20 :
C
C : 20 :
20 :
C
C
20 :
A : 20 :
..A : 20 :
A : 20 :
C : 24 :
C : 24 :
C : 24 :
A • 24 :
A : 24 :
6 :
17 :
18 :
19 :
20 :
21
7 :
8:
9 :
10 :
11 :
12
22 :
23 :
24 :
13 :
14 :
15 :
25 :
26 :
27 :
A :
24 :
Values of a and b in equation
w = atb
.0029
.0023
.0033
•
.0031
:
•.
.0069
:
.0049
:
.0032
:
.0021
.0038
.0037
.0039
:
:
:.
:
.0039
:
.0031
.0049
.0059
.0041
:
.0025
.0043
.0035
.0043
.0034
.0037
.0033
:.
:
:.
:
:
•
:
.208
.241
.196
:
:
:
.209
.130
•
.160
. 174
:
•.
.297
.201
.210
.201
.280
••
:
:
•.
2
.0035 :
.0025 :
.0029 •
.0035 :
.0033 :
.004o :
.0039 :
.0040 :
.0038 :
.223
.280
.172
.120
.
:
.142
.244
.174
:
:
•.
.220
.235
:
:
:
:
•
Report No. 1324
.0028 :
.0029 :
.0039 :
.207
.285
.0028
7
5 : .0028
.0028
7
.0028
8
.0027
.0026 :
.0031 :
.0035 :
.0039 :
.0033 :
.0034 :
.0040
.0044 •.
..0028
.0028
.0028
5 :
7
7 :
5 : .O00386,
.0036
6
7 : .0036
6 : .0036
5 : .0036
.0036
4
5 : .0036
7 : .0036
.0036
8
.0038
7
8 : .0038
7 : .0038
7. : .0038
7 : .0038
.0038
7
•
SUBJECT LISTS OF PUBLICATIONS ISSUED BY THE
FOREST PRODUCTS LABORATORY
The following are obtainable free on request from the Director, Forest Products
Laboratory, Madison 5, Wisconsin:
List of publications on
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List of publications on
Fungus Defects in. Forest
Products and Decay in Trees
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List of publications on
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and Veneer
List of publications on
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List of publications on
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List of publications on
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Laminates, and Wood-Base
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List of publications on
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List of publications on
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Structural Uses of Wood
and Wood Products
List of publications on
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Partial list of publications
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Engineers, and Retail
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Partial list of publications
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Woodworkers and Teachers of
Woodehop Practice
Note:
Since Forest Products Laboratory publications are so varied in subject
no single list is issued. Instead a list is made up for each Laboratory
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